Analytical estimates of strutural behavior

Analytical

Estimates of

Structural

Behavior

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Analytical

Estimates of

Structural

Behavior

Clive L. Dym | Harry E. Williams

CRC Press is an imprint of the

Taylor & Francis Group, an informa business

Boca Raton London New York

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CRC Press

Taylor & Francis Group

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© 2012 by Taylor & Francis Group, LLC

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Version Date: 20111007

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Library of Congress Cataloging‑in‑Publication Data

Dym, Clive L.

Analytical estimates of structural behavior / Clive L. Dym and Harry E. Williams.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-4398-7091-4 (acid-free paper)

1. Elasticity. 2. Structural dynamics. 3. Structural analysis (Engineering) I.

Williams, Harry E. (Harry Edwin), 1930- II. Title.

TA418.D88 2012

624.1’76--dc23 2011035549

Visit the Taylor & Francis Web site at

http://www.taylorandfrancis.com

and the CRC Press Web site at

http://www.crcpress.com

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In the spirit of “standing on the shoulders of giants,” we dedicate this

book to our mentors (and the institutions at which we began to learn)

Anthony E. Armenàkas (Cooper Union)

Joseph Kempner (Brooklyn Polytechnic Institute)

Nicholas J. Hoff (Stanford University)

and

Richard M. Hermes (Santa Clara University)

Julian D. Cole (California Institute of Technology)

George W. Housner (California Institute of Technology)

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vii

Contents

Preface......................................................................................................................xi

Authors ...................................................................................................................xv

1 Mathematical Modeling for Structural Analysis.....................................1

Summary...........................................................................................................1

Principles of Mathematical Modeling...........................................................2

Abstraction and Scale (I).................................................................................5

Abstraction, Scaling, and Lumped Elements..........................................5

Dimensional Consistency and Dimensional Analysis...............................7

Dimensions and Units................................................................................7

Dimensionally Homogeneous Equations and

Unit-Specific Formulas ...............................................................................8

The Basic Method of Dimensional Analysis ...........................................9

The Buckingham Pi Theorem of Dimensional Analysis.....................12

Abstraction and Scale (II) ............................................................................. 14

Geometric Scaling..................................................................................... 15

Scale in Equations: Size and Limits........................................................ 19

Consequences of Choosing a Scale.........................................................22

Scaling and the Design of Experiments.................................................22

Notes on Approximating: Dimensions and Numbers .............................26

The Assumption of Linear Behavior...........................................................29

Linearity and Geometric Scaling............................................................30

Conservation and Balance Principles ......................................................... 31

Conclusions..................................................................................................... 32

Bibliography....................................................................................................33

2 Structural Models and Structural Modeling ..........................................35

Summary.........................................................................................................35

Bars, Beams, Arches, Frames, and Shells ...................................................35

One-Dimensional Structural Elements.......................................................36

Stress Resultants for One-Dimensional Structural Elements.............38

Two-Dimensional Structural Elements ......................................................40

Modeling Structural Supports ..................................................................... 41

Indeterminacy, Redundancy, Load Paths, and Stability ..........................44

Counting Degrees of Indeterminacy .....................................................45

Indeterminacy and Redundancy Matter...............................................46

Important Aspects of Structural Stability .............................................49

Modeling Structural Loading ...................................................................... 51

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viii Contents

Modeling Structural Materials ....................................................................53

Idealization and Discretization in Structural Modeling..........................55

Bibliography....................................................................................................56

3 Exploring Intuition: Beams, Trusses, and Cylinders ............................ 57

Summary......................................................................................................... 57

Introduction.................................................................................................... 57

Engineering Beams: The Two-Dimensional Model..................................58

Reasoning Intuitively about Engineering Beams .....................................68

Trusses as Beams ........................................................................................... 76

Pressurized Circular Cylinders: A Two-Dimensional Model.................80

Conclusions.....................................................................................................85

Bibliography....................................................................................................86

4 Estimating Stresses and Displacements in Arches ...............................89

Summary.........................................................................................................89

Introduction....................................................................................................89

Normal and Bending Stresses in Transversely Loaded Arches ............. 91

Arches under Centrally Applied, “Dead” Loading..................................96

Shallow Arches under Centrally Directed, “Dead” Loading................ 103

Arches under Gravitational Loading........................................................ 105

Interpreting Arch Behavior........................................................................ 107

Shallow Curved Beams under End Loading........................................... 114

Interpreting Curved Beam Behavior ........................................................ 117

Conclusions................................................................................................... 120

Bibliography.................................................................................................. 121

5 Estimating the Behavior of Coupled Discrete Systems......................123

Summary.......................................................................................................123

Introduction..................................................................................................123

Extending the Castigliano Theorems to Discrete Systems....................125

Formally Proving the Castigliano Theorem Extensions........................ 131

Rayleigh Quotients for Discrete Systems ................................................. 134

Conclusions................................................................................................... 139

Bibliography.................................................................................................. 139

6 Buildings Modeling as Coupled Beams: Static Response

Estimates ...................................................................................................... 141

Summary....................................................................................................... 141

Introduction.................................................................................................. 141

Coupled Timoshenko Beams: Exact Solutions ........................................ 145

Coupled Timoshenko Beams: Castigliano Estimates (I)........................ 150

Validating Castigliano-Based Deflection Estimates ............................... 156

Coupled Timoshenko Beams: Castigliano Estimates (II)....................... 159

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Contents ix

Remarks on Modeling Buildings as Coupled-Beam Systems ............... 162

Conclusions................................................................................................... 164

Appendix A: Exact Solution for Coupled Timoshenko Beams ............. 165

Appendix B: The Coupled Euler–Shear System as a Limit.................... 167

Appendix C: Special Cases for Two Timoshenko Cantilevers .............. 168

Bibliography.................................................................................................. 171

7 Buildings Modeled as Coupled Beams: Natural Frequency

Estimates ...................................................................................................... 173

Summary....................................................................................................... 173

Introduction.................................................................................................. 173

Rayleigh Quotients for Elementary Beams .............................................. 174

Beams and Models of Buildings................................................................ 180

Frequency–Height Dependence in Coupled Two-Beam Models .......... 182

Frequency-Height Dependence in Timoshenko Beams......................... 185

Comparing Frequencies for Coupled Two-Beam Models...................... 188

Conclusions................................................................................................... 192

Bibliography.................................................................................................. 193

Index ..................................................................................................................... 197

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xi

Preface

We intend this book to explicitly return the notion of modeling to the anal￾ysis of structures by presenting an integrated approach to modeling and

estimating structural behavior. The advent of computer-based approaches

to structural analysis and design over the last 50 years has only accentuated

the need for structural engineers to recognize that we are dealing with mod￾els of structures, rather than with the actual structures. Further, as tempt￾ing as it is to run innumerable computer simulations, closed-form estimates

can be effectively used to guide and to check numerical results, as well as

to confirm in clear terms physical insights and intuitions. What is truly

remarkable is that the way of thinking about structures and their models

that we propose is rooted in classic elementary elasticity: It depends less

on advanced mathematical techniques and far more on thinking about the

dimensions and magnitudes of the underlying physics.

A second reason for this book is our concern with traditional classroom

approaches to structural analysis. Most introductory textbooks on struc￾tural analysis convey the subject as a collection of seemingly unrelated

tools available to handle a set of relatively specific problem types. A major

divide on the problem-type axis is the distinction drawn between struc￾tures that are statically determinate and those that are not. While this also

logically conforms to a presentation in an order that reflects the respec￾tive degree of difficulty of application, it is often not seen by students as a

coherent view of the discipline. Perhaps reflecting a long-standing split in

professional affiliations, the classical approaches to structural analysis are

often presented as a field entirely distinct from its logical underpinnings in

mechanics, especially applied mechanics.

Finally, as noted before, the advent of the computer and its ubiquitous

use in the classroom and in the design office has led structural engineer￾ing faculty to include elementary computer programs within a shrinking

structural curriculum. Thus, students seem to spend more time and effort

generating numbers, with less time and effort spent on understanding what

meaning—if any—to attach to the numbers that are generated with these

programs. This tendency has only strengthened as computers have become

still more powerful. Still more unfortunate is that this approach empha￾sizes another growing dissonance in the education of engineering students:

Problems in structural behavior and response continue to be formulated

largely in mathematical terms, while solutions are increasingly sought with

computer programs.

We have based this book on the premise that it is now even more impor￾tant to understand basic structural modeling, with strong emphasis on

understanding behavior and interpreting results in terms of the limitations

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xii Preface

of the models being applied. In fact, we would argue that the generation of

numerical analyses for particular cases is, in the “real world,” increasingly

a task performed by technicians or entry-level engineers, rather than by

seasoned professional engineers. As numerical analysis becomes both more

common and significantly easier, those structural analysts and designers

who know which calculations to perform, how to validate and interpret those

calculations, and what the subsequent results mean will be the most highly

regarded engineers. The knowledge needed to do these tasks can often be

encapsulated and illustrated with the ability to obtain and properly use

analytical, closed-form estimates—or, in other words, the ability to obtain

and properly use “back of the envelope” models and formulas.

We note that it is more than the outline of topics that sets apart this book

from others. That outline, to be described immediately, is not what we would

expect to find in a first course in structural analysis. In fact, much of what we

have included in Chapters 3–7 derives from articles we have published in the

various research journals (see the references and bibliographies at the end

of each chapter). The common theme of these articles and of Chapters 3–7 is

the development of effective analytical estimates of the responses of certain

structural models. So, we hope to stretch the mold of traditional approaches

to structural analysis—and especially how our colleagues teach structural

analysis—to emphasize and more explicitly describe the modeling process,

and thus present a more conscious view of estimating and assessing struc￾tural response.

We finally note that while this book is not intended as a text for a first

course in structural analysis, we certainly think it is accessible to advanced

undergraduates as well as to graduate students and practitioners. It does

not require deep knowledge of advanced structural mechanics models or

techniques:

• We use the principle of minimum total potential energy to derive

governing equations and boundary conditions, but those equations

can be derived in other ways or even simply accepted.

• We introduce extensions of the Castigliano theorems and Rayleigh

quotients for discrete systems, laying a foundation for applying

them to continuous systems.

The mathematical skills that will be exercised are more about applying tech￾niques of dimensional analysis, reasoning about physical dimensions, and

reasoning about the relative sizes of mathematical terms and using appro￾priate expansions to determine limits and limiting behavior.

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Preface xiii

Organization

This book is organized as follows. In Chapter 1 we outline some important

principles and techniques of mathematical modeling, including dimensional

analysis, scaling, linearity, and balance and conservation laws. In Chapter 2

we review basic structural models, including structural supports and mate￾rials, as well as some general considerations of load paths, redundancy,

determinacy, and stability. We also review there the concept of idealization,

and we complete the chapter by bringing discretization under the modeling

umbrella as well.

In Chapter 3 we use subsets of two-dimensional elasticity theory to recon￾sider two classic structural mechanics problems so as to explore how we

develop and express physical intuition. First, we rederive the traditional

fourth-order Euler–Bernoulli beam equation and boundary conditions and

then use these results to estimate ranges of validity for beam models. Intuition

issues emerge as we interpret both boundary conditions, the beam’s physi￾cal parameters, and the nature of the loading—in particular, the transition

from sets of concentrated loads to a uniform load. We illustrate how planar

truss configurations behave as beams and use two-dimensional elasticity to

derive another classical problem, the static response of pressure-loaded cyl￾inders, and show how our physical intuitions can lead us astray.

In Chapter 4 we demonstrate how the behavior of arches under lateral load

can be tracked as it varies from beam behavior at small values of an arch

parameter (i.e., arches with very small rises) to purely compressive arch behav￾ior when the arch parameter is large (i.e., for large arch rises). It is also shown

that the behavior “flips” when the load applied is axial, rather than lateral.

In Chapter 5 we introduce two methods of analyzing coupled discrete sys￾tems, in part to lay a foundation for their application to continuous systems

in our two final chapters, and in part just to ensure a common background

for readers who may not be familiar with either or both of the techniques

described. First, we describe recently developed extensions of Castigliano’s

theorems, and then we introduce Rayleigh’s quotient for estimating the fun￾damental frequencies of coupled spring-mass oscillators. Then, in Chapter 6

we apply the extension of Castigliano’s second theorem to derive simple, yet

quite accurate estimates of the transverse displacements of structures mod￾eled in terms of coupled Timoshenko beams (e.g., tall buildings). Finally, in

a similar vein, in Chapter 7 we use Rayleigh quotients to analyze the dimen￾sional behavior of and calculate numerical values of fundamental frequen￾cies of structures modeled in terms of Euler–Bernoulli, Timoshenko, and

coupled-beam systems (e.g., again, potential models of tall buildings).

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xv

Authors

Clive L. Dym is Fletcher Jones Professor of Engineering Design and director

of the Center for Design Education at Harvey Mudd College. After receiv￾ing his PhD from Stanford University, Dr. Dym held appointments at the

University of Buffalo; the Institute for Defense Analyses; Carnegie Mellon

University; Bolt, Beranek and Newman; and the University of Massachusetts

at Amherst. He has held visiting appointments at the TECHNION-Israel

Institute of Technology, the Institute for Sound and Vibration Research

at Southampton, Stanford, Xerox PARC, Carnegie Mellon, Northwestern,

USC, and the Singapore University of Technology and Design. Dr. Dym

has authored or coauthored more than a dozen books and 90 refereed

journal articles, was founding editor of the journal Artificial Intelligence

for Engineering Design, Analysis, and Manufacturing, and has served on the

editorial boards of several other journals, including the ASME’s Journal

of Mechanical Design. His primary interests are in engineering design and

structural mechanics.

Dr. Dym is a fellow of the Acoustical Society of America, the American

Society of Mechanical Engineers, the American Society of Civil Engineers,

and the American Society for Engineering Education, and is a member of

the American Academy of Mechanics. Dr. Dym’s awards include the Walter

L. Huber Research Prize (ASCE, 1980), the Western Electric Fund Award

(ASEE, 1983), the Boeing Outstanding Educator Award (first runner-up,

2001), the Fred Merryfield Design Award (ASEE, 2002), the Joel and Ruth

Spira Outstanding Design Educator Award (ASME, 2004), the Archie Higdon

Distinguished Educator Award (Mechanics Division, ASEE, 2006), and the

Bernard M. Gordon Prize for Innovation in Engineering and Technology

Education (NAE, 2012; co-winner).

Harry E. Williams is professor emeritus of engineering at Harvey Mudd

College. After receiving his PhD from the California Institute of Technology,

Dr. Williams joined the research staff of the Jet Propulsion Laboratory

and then joined Harvey Mudd College as one of the founding faculty of

its engineering program, serving there for 40 years. He has also been a

Fulbright fellow at the University of Manchester, a liaison scientist for the

Office of Naval Research, and a consultant to General Dynamics, Teledyne

Microelectronics, the Naval Weapons Center, Aerojet-General, and the

Boeing Company. He has published widely over the years, reflecting his

interests in fluid mechanics, thermoelasticity, and the mechanics of solids

and structures.

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1

1

Mathematical Modeling for

Structural Analysis

Summary

The dictionary defines a model as “a miniature representation of something;

a pattern of something to be made; an example for imitation or emulation;

a description or analogy used to help visualize something (e.g., an atom)

that cannot be directly observed; a system of postulates, data and inferences

presented as a mathematical description of an entity or state of affairs.” This

definition suggests that modeling is an activity, a cognitive activity in which

one thinks about and makes models to describe how devices or objects of

interest behave. Thus, it is important to remember that when we describe

or formulate a problem in words, draw a sketch (e.g., a free-body diagram),

write down or derive a formula, and crank through to get some numbers, we

are modeling something. In each of these activities we are formulating and

representing a model of the problem in a modeling language. And as we go

from words to pictures to formulas to numbers, we must be sure that we are

translating our problem correctly and consistently. We have to maintain our

assumptions, and at the right level of detail.

Since there are many ways in which devices and behaviors can be

described—words, drawings or sketches, physical models, computer pro￾grams, or mathematical formulas—it is worth refining the foregoing dic￾tionary definition to define a mathematical model as a “representation in

mathematical terms” of the behavior of real devices and objects. Our pri￾mary modeling language is mathematics, so we must be able to translate

fluently into and from mathematics.

Scientists use mathematical models to describe observed behavior or results,

explain why that behavior and those results occurred as they did, and predict

future behaviors or results that are as yet unseen or unmeasured. Engineers

use mathematical models to describe and analyze objects and devices

in order to predict their behavior because they are interested in designing

devices and processes and systems. Design is a consequential activity for

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2 Analytical Estimates of Structural Behavior

engineers because every new airplane or building, for example, represents a

model-based prediction that the plane will fly and the building stand with￾out dire, unanticipated consequences. Further, as practicing engineers, we

must always remember that we are dealing with models of a problem—

models of reality. Thus, if our results do not match experimental data or intu￾itive expectations, we may well have a model that is simply wrong. So it

is especially important in engineering to ask: How are such mathematical

models or representations created? How are they validated? How are they

used? Is their use limited and, if so, how?

To answer these and related questions, this chapter first sets out some

basic principles of mathematical modeling and then goes on to describe

briefly:

• abstraction and scaling

• dimensional consistency and dimensional analysis

• conservation and balance laws

• the assumption of linear behavior

Principles of Mathematical Modeling

Mathematical modeling is a principled activity that has principles behind it

as well as methods that can be successfully applied. The principles are over￾arching or metaprinciples that are almost philosophical in nature, and they

can be phrased as questions (and answers) about modeling tasks we need

to perform and their purposes. That is, builders of mathematical (and other

types of) models must identify

a. The need for the model: Why is this being done?

b. The data sought: What information is being sought?

c. The available relevant data: What is known (i.e., What is given?)

d. The circumstances that apply: What can be assumed?

e. The governing physical principles: How should this model be viewed?

f. The equations that will be used, the calculations that will be made,

and the answers that will result: What will the model predict?

g. The tests to be made to validate the model and ensure its consistency

with its principles and assumptions: Are the predictions valid?

h. The tests to be made to verify the model and ensure its usefulness

in terms of the initial reason it was done: Can the predictions be

verified?

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Mathematical Modeling for Structural Analysis 3

i. Parameter values that are not adequately known, variables that

should have been included, and/or assumptions that could be

removed (i.e., can an iterative “model-validate-verify-improve￾predict” loop be implemented? Can the model be improved?)

j. What will be done with the model: How will the model be used?

These identified tasks and questions can also be visually portrayed (see

Figure 1.1).

Object or System

(To be modeled)

Why is this being done?

What information is being sought?

Model

Variables and Parameters

Can the model be improved?

What is given?

What can be assumed?

How should this model be viewed?

What will this model predict?

Model Predictions Test

Valid, Accepted Predictions

Are the predictions valid?

Can the predictions be verified?

Figure 1.1

A graphical overview of mathematical modeling shows how the questions asked during a prin￾cipled approach to model building relate to the development of that model. (Dym, C. L. 2004.

Principles of Mathematical Modeling, 2nd ed. By permission of Elsevier Academic Press.)

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